#include <cmath>
#include <cstdint>
#include <ctime>
#include <functional>
#include <iostream>
#include <random>
#include <vector>

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Namespaces
namespace	math
	for assert

namespace	monte_carlo
	Functions for the Monte Carlo Integration implementation.

Typedefs
using	math::monte_carlo::Function = std::function< double(double &)>

Functions
std::vector< double >	math::monte_carlo::generate_samples (const double &start_point, const Function &pdf, const uint32_t &num_samples, const uint32_t &discard=100000)
	short-hand for std::functions used in this implementation More...

double	math::monte_carlo::integral_monte_carlo (const double &start_point, const Function &function, const Function &pdf, const uint32_t &num_samples=1000000)
	Compute an approximation of an integral using Monte Carlo integration. More...

static void	test ()
	Self-test implementations. More...

int	main ()
	Main function. More...

Detailed Description

Monte Carlo Integration

In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integrals.

This implementation supports arbitrary pdfs. These pdfs are sampled using the Metropolis-Hastings algorithm. This can be swapped out by every other sampling techniques for example the inverse method. Metropolis-Hastings was chosen because it is the most general and can also be extended for a higher dimensional sampling space.

Author: Domenic Zingsheim

Function Documentation

◆ generate_samples()

std::vector< double > math::monte_carlo::generate_samples	(	const double &	start_point,
		const Function &	pdf,
		const uint32_t &	num_samples,
		const uint32_t &	discard = `100000`
	)

short-hand for std::functions used in this implementation

Generate samples according to some pdf

This function uses Metropolis-Hastings to generate random numbers. It generates a sequence of random numbers by using a markov chain. Therefore, we need to define a start_point and the number of samples we want to generate. Because the first samples generated by the markov chain may not be distributed according to the given pdf, one can specify how many samples should be discarded before storing samples.

Parameters

start_point	The starting point of the markov chain
pdf	The pdf to sample
num_samples	The number of samples to generate
discard	How many samples should be discarded at the start

Returns: A vector of size num_samples with samples distributed according to the pdf

                                                                                                                                                  {
    std::vector<double> samples;
    samples.reserve(num_samples);
 
    double x_t = start_point;
 
    std::default_random_engine generator;
    std::uniform_real_distribution<double> uniform(0.0, 1.0);
    std::normal_distribution<double> normal(0.0, 1.0);
    generator.seed(time(nullptr));
 
    for(uint32_t t = 0; t < num_samples + discard; ++t) {
        // Generate a new proposal according to some mutation strategy.
        // This is arbitrary and can be swapped.
        double x_dash = normal(generator) + x_t;
        double acceptance_probability = std::min(pdf(x_dash)/pdf(x_t), 1.0);
        double u = uniform(generator);
 
        // Accept "new state" according to the acceptance_probability
        if(u <= acceptance_probability) {
            x_t = x_dash;
        }
 
        if(t >= discard) {
            samples.push_back(x_t);
        }
    }
 
    return samples;
}

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◆ integral_monte_carlo()

double math::monte_carlo::integral_monte_carlo	(	const double &	start_point,
		const Function &	function,
		const Function &	pdf,
		const uint32_t &	num_samples = `1000000`
	)

Compute an approximation of an integral using Monte Carlo integration.

The integration domain [a,b] is given by the pdf. The pdf has to fulfill the following conditions: 1) for all x \in [a,b] : p(x) > 0 2) for all x \not\in [a,b] : p(x) = 0 3) \int_a^b p(x) dx = 1

Parameters

start_point	The start point of the Markov Chain (see generate_samples)
function	The function to integrate
pdf	The pdf to sample
num_samples	The number of samples used to approximate the integral

Returns: The approximation of the integral according to 1/N \sum_{i}^N f(x_i) / p(x_i)

                                                                                                                                             {
    double integral = 0.0;
    std::vector<double> samples = generate_samples(start_point, pdf, num_samples);
 
    for(double sample : samples) {
        integral += function(sample) / pdf(sample);
    }
 
    return integral / static_cast<double>(samples.size());
}

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◆ main()

int main ( void )

Main function.

Returns: 0 on exit

           {
    test();  // run self-test implementations
    return 0;
}

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◆ test()

static void test ( )

static

Self-test implementations.

Returns: void

                   {
    std::cout << "Disclaimer: Because this is a randomized algorithm," << std::endl;
    std::cout << "it may happen that singular samples deviate from the true result." << std::endl << std::endl;;
 
    math::monte_carlo::Function f;
    math::monte_carlo::Function pdf;
    double integral = 0;
    double lower_bound = 0, upper_bound = 0;
 
    /* \int_{-2}^{2} -x^2 + 4 dx */
    f = [&](double& x) {
        return -x*x + 4.0;
    };
 
    lower_bound = -2.0;
    upper_bound = 2.0;
    pdf = [&](double& x) {
        if(x >= lower_bound && x <= -1.0) {
            return 0.1;
        }
        if(x <= upper_bound && x >= 1.0) {
            return 0.1;
        }
        if(x > -1.0 && x < 1.0) {
            return 0.4;
        }
        return 0.0;
    };
 
    integral = math::monte_carlo::integral_monte_carlo((upper_bound - lower_bound) / 2.0, f, pdf);
 
    std::cout << "This number should be close to 10.666666: " << integral << std::endl;
 
    /* \int_{0}^{1} e^x dx */
    f = [&](double& x) {
        return std::exp(x);
    };
 
    lower_bound = 0.0;
    upper_bound = 1.0;
    pdf = [&](double& x) {
        if(x >= lower_bound && x <= 0.2) {
            return 0.1;
        }
        if(x > 0.2 && x <= 0.4) {
            return 0.4;
        }
        if(x > 0.4 && x < upper_bound) {
            return 1.5;
        }
        return 0.0;
    };
 
    integral = math::monte_carlo::integral_monte_carlo((upper_bound - lower_bound) / 2.0, f, pdf);
 
    std::cout << "This number should be close to 1.7182818: " << integral << std::endl;
 
    /* \int_{-\infty}^{\infty} sinc(x) dx, sinc(x) = sin(pi * x) / (pi * x)
       This is a difficult integral because of its infinite domain.
       Therefore, it may deviate largely from the expected result.
    */
    f = [&](double& x) {
        return std::sin(M_PI * x) / (M_PI * x);
    };
 
    pdf = [&](double& x) {
        return 1.0 / std::sqrt(2.0 * M_PI) * std::exp(-x * x / 2.0);
    };
 
    integral = math::monte_carlo::integral_monte_carlo(0.0, f, pdf, 10000000);
 
    std::cout << "This number should be close to 1.0: " << integral << std::endl;
}

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Namespaces

Typedefs

Functions

Detailed Description

Function Documentation

◆ generate_samples()

◆ integral_monte_carlo()

◆ main()

◆ test()